How to evenly distribute tiles on Cone
Theory
I don't think you have a chance to get there with this single texture and your mesh topology.
The only reasonable solution that came to my mind is to use some radial texture, like one based on fibonacci pattern.

Unwrapped from top view.
Pattern
To create such texture use this Brecht's script
(copy&amp;paste into blender's Script Editor and Run Alt+P)
import bpy
import bmesh
import math

n = 100 # number of points
c = 0.1 # scale factor

mesh = bpy.data.meshes.new(name=&quot;Spiral&quot;)
bm = bmesh.new()

for i in range(0, n):
    theta = i * math.radians(137.5)
    r = c * math.sqrt(i)
    bm.verts.new((math.cos(theta) * r, math.sin(theta) * r, 0.0))

bm.to_mesh(mesh)
mesh.update()

from bpy_extras import object_utils
object_utils.object_data_add(bpy.context, mesh)

It creates vertices pattern. At row 5 of the script you can change number of vertices.

Model
Add Plane and model one Shingle, Object has to be oriented in positive Y direction so bottom part is on top. Rotate it on X axis a bit upward with shingles end so when they will repeated will be on top of other. For this quick Shingle I added Solidify and Bevel modifier.

Select Shingle and than pattern object and parent Ctrl+P them. Go to pattern's Object Properties &gt; Instances &gt; Vertices. Enable Align to Vertex Normal to get desired rotation.


Texture
To be able bake Normal map of this model you have to search for Make Instances Real, Join Ctrl+J them. Add another Plane above, add a new material and add Image node, click new, and keep this node selected (active). Go to Render Properties, choose Cycles engine, at bottom go to Bake &gt; Bake Type &gt; Normal &gt; enable Selected to Active. Select joined Shingles than Plane and press BAKE. Save image as OpenEXR type of file.

Note: to create Bump Map - position camera above object and under Render Passes enable Depth pass or Mist pass. After rendering go to Compositor and tweak output of one of this pass. Or you can create a simple B&amp;W material with Blend texture on Z axis for the same result. You can find a lot of answers here I guess.
Edit:
I just noticed ... baked Ambient Oclussion worked much better in this case. Result is similar to Depth. And you could already notice, unwrap from top view applies texture stretched vertically, so its good to compensate this by creating shingle a bit wider horizontal, so after texture is applied it looks in proportions.


Render doesn't use Displacement modifier, but Adaptive Subdivision (aka Microdisplacement) that needs a just a few faces for cone and beautiful details :)